FLUVIAL BEDFORMS

1
CHAPTER 8 FLUVIAL BEDFORMS
The interaction of flow and sediment transport
often creates bedforms such as ripples, dunes,
antidunes, and bars. These bedforms in turn can
interact with the flow to modify the rate of
sediment transport.
Dunes in the North Loup River, Nebraska, USA
image courtesy D. Mohrig
2
TOUR OF BEDFORMS IN RIVERS RIPPLES
Ripples are characteristic of a) very low
transport rates in b) rivers with sediment size D
less than about 0.6 mm. Typical wavelengths ?
are on the order of 10s of cm and and wave
heights ?? are on the order of cm. Ripples
migrate downstream and are asymmetric with a
gentle stoss (upstream) side and a steep lee
(downstream side). Ripples do not interact with
the water surface.
View of the Rum River, Minnesota USA
Ripples in the Rum River, Minnesota USA at very
low flow ? 10 - 20 cm.
3
TOUR OF BEDFORMS IN RIVERS DUNES
Dunes are the most common bedforms in sand-bed
rivers they can also occur in gravel-bed rivers.
Wavelength ? can range up to 100s of m, and
wave height ?? can range up to 5 m or more in
large rivers. Dunes are usually asymmetric, with
a gentle stoss (upstream) side and a steep lee
(downstream) side. They are
characteristic of subcritical flow (Fr
sufficiently below 1). Dunes migrate
downstream. They interact weakly with the water
surface, such that the flow accelerates over the
crests, where water surface elevation is slightly
reduced. (That is, the water surface is out of
phase with the bed.)
Dunes in the North Loup River, Nebraska USA. Two
people are circled for scale. Image courtesy D.
Mohrig.
4
TOUR OF BEDFORMS IN RIVERS DUNE MIGRATION
Double-click on the image to see the video video
courtesy D. Mohrig.
rte-bookmohrigloup.mpg to run without relinking,
download to same folder as PowerPoint
presentations.
5
TOUR OF BEDFORMS IN RIVERS ANTIDUNES
Antidunes occur in rivers with sufficiently high
(but not necessarily supercritical) Froude
numbers. They can occur in sand-bed and
gravel-bed rivers. The most common type of
antidune migrates upstream, and shows little
asymmetry. The water surface is strongly in
phase with the bed. A train of symmetrical
surface waves is usually indicative of the
presence of antidunes.
Trains of surface waves indicating the presence
of antidunes in braided channels of the tailings
basin of the Hibbing Taconite Mine, Minnesota,
USA. Flow is from top to bottom.
6
TOUR OF BEDFORMS IN RIVERS CYCLIC STEPS (CHUTE
AND POOL TRAINS)
Trains of cyclic steps occur in very steep flows
with supercritical Froude numbers. They are
long-wave relatives of antidunes. The steps are
delineated by hydraulic jumps (immediately
downstream of which the flow is locally
subcritical). The steps migrate upstream. These
features are also called chute-and-pool
topography.
Train of cyclic steps in a small laboratory
channel at St. Anthony Falls Laboratory. The
water has been dyed to aid visualization two
hydraulic jumps can be seen in the figure.
7
TOUR OF BEDFORMS IN RIVERS CYCLIC STEPS (contd.)
Cyclic steps form in the field when slopes are
steep, the flow is supercritical and there is a
plethora of sediment.
jumps
flow
Trains of cyclic steps in a coastal outflow
channel on a beach in Calais, France. Image
courtesy H. Capart.
8
TOUR OF BEDFORMS IN RIVERS ALTERNATE BARS
Alternate bars occur in rivers with sufficiently
large (gt 12), but not too large width-depth
ratio B/H. Alternate bars migrate downstream,
and often have relatively sharp fronts. They are
often precursors to meandering. Alternate bars
may coexist with dunes and/or antidunes.
Alternate bars in the Naka River, an artificially
straightened river in Japan. Image courtesy S.
Ikeda.
9
TOUR OF BEDFORMS IN RIVERS MULTIPLE-ROW LINGUOID
BARS
Multiple-row bars (linguoid bars) occur when the
width-depth ratio B/H is even larger than that
for alternate bars. These bars migrate
downstream. They may co-exist with dunes or
antidunes.
Plan view of superimposed linguoid bars and dunes
in the North Loup River, Nebraska USA. Image
courtesy D. Mohrig. Flow is from left to right.
10
BEDFORMS IN THE LABORATORY AND FIELD DUNES
Dunes in a flume in Tsukuba University, Japan
flow turned off. Image courtesy H. Ikeda.
Dunes on an exposed point bar in the meandering
Fly River, Papua New Guinea
11
Rhine River, Switzerland
BEDFORMS IN THE LABORATORY AND FIELD ALTERNATE
BARS
Alternate bars in a flume in Tsukuba University,
Japan flow turned low. Image courtesy H. Ikeda.
Alternate bars in the Rhine River between
Switzerland and Lichtenstein. Image courtesy M.
Jaeggi.
12
BEDFORMS IN THE LABORATORY AND FIELD
MULTIPLE-ROW (LINGUOID) BARS
Linguoid bars in a flume in Tsukuba University,
Japan flow turned off. Image courtesy H. Ikeda.
Linguoid bars in the Fuefuki River, Japan. Image
courtesy S. Ikeda.
13
Ohau River, New Zealand
WHEN THE FLOW IS INSUFFICIENT TO COVER THE BED,
THE RIVER MAY DISPLAY A BRAIDED PLANFORM
Braiding in a flume in Tsukuba University, Japan
flow turned low. Image courtesy H. Ikeda.
Braiding in the Ohau River, New Zealand. Image
courtesy P. Mosley.
14
RIPPLES
Ripples are small-scale bedforms that migrate
downstream and show a characteristic asymmetry,
with a gentle stoss face and a steep lee face.
Ripples require the existence of a reasonably
well-defined viscous sublayer in order to form.
In rivers, a viscous sublayer can exist only when
the flow is very slow and well below flood
conditions. Because of the viscous sublayer,
ripples do not interact with the water
surface. Engelund and Hansen (1967) have
suggested the following condition for ripple
formation D ? ?v, where ??v 11.6 ?/u denotes
the thickness of the viscous sublayer (Chapter
6). This relation can be rearranged to yield
the threshold condition
where
The above relation can be solved with the
modified Brownlie relation of Chapter 6 to yield
a maximum value of Rep for ripple formation. The
value so obtained is 91, corresponding to a grain
size of 0.8 mm with ? 0.01 cm2/s and R 1.65.
In practice, ripples are observed only for D lt
0.6 mm. Ripples can coexist with dunes.
15
SHIELDS DIAGRAM WITH CRITERION FOR RIPPLES
16
DEFINITION OF DUNES AND ANTIDUNES
Dunes are 1D (or quasi-1D) bedforms for which the
water surface fluctuations are approximately out
of phase with the bed fluctuations. That is, the
water surface is high where the bed is low and
vice versa. As is shown below dunes migrate
downstream. Antidunes are 1D (or quasi-1D)
bedforms for which the water surface fluctuations
are approximately in phase with the bed
fluctuations. That is, the water surface is high
where the bed is high and vice versa. As shown
below, most antidunes migrate upstream, but there
is a regime within which they can migrate
downstream.
17
RESPONSE OF FLOW TO BED UNDULATIONS INVISCID
SHALLOW-WATER FORMULATION FOR 1D BEDFORMS
Steady, uniform flow over a flat erodible bed
(base flow no bedforms) has flow depth Ho and
flow velocity Uo qw/Ho. Unperturbed bed
elevation is at ? 0. The bed is then given a
slight wavy perturbation of the form where ?
ltlt Ho denotes the amplitude of the perturbation
and ? denotes the wavelength of the perturbation.
How does the flow and water surface respond to
such a perturbation?
18
RESPONSE OF FLOW TO BED UNDULATIONS INVISCID
SHALLOW-WATER FORMULATION FOR 1D BEDFORMS contd.
Consider inviscid (frictionless) steady 1D
shallow water flow over an undulating bed. The
St. Venant shallow water equations simplify as
follows
The equation in the box can be made dimensionless
using the depth Ho of the base flow
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RESPONSE OF FLOW TO BED UNDULATIONS LINEAR
INVISCID SHALLOW-WATER FORMULATION FOR 1D BEDFORMS
Solving for the variation in flow depth, The
variation in water surface elevation is given as
Now the bed perturbation can be represented in
dimensionless form as follows Here denotes
the dimensionless amplitude of the bed
perturbation and k denotes the dimensionless
wavenumber of the bed perturbation. We further
write the response of the depth and water surface
elevation to the perturbation as where
denotes the dimensionless amplitude of the
response of depth to the bed perturbation, and
denotes the corresponding dimensionless
response in water surface elevation.
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RESPONSE OF FLOW TO BED UNDULATIONS LINEAR
INVISCID SHALLOW-WATER FORMULATION FOR 1D
BEDFORMS contd.
Now as long as ltlt 1, With this
approximation, substituting into gives the
results
and
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SHALLOW-WATER RESPONSE OF WATER SURFACE TO BED
PERTURBATION
When Fro lt 1, and the water
surface perturbation is out of phase with the bed
perturbation the water surface is low
where the bed is high and the water
surface is high where the bed is low. According
to long wave theory, then, dunes can occur in
subcritical flow (Fro lt 1)
When Fr0 gt 1, and the depth
perturbation is in phase with the bed
perturbation the water surface is high
where the bed is high and the water
surface is low where the bed is low. According
to long wave theory, then, antidunes can occur in
supercritical flow (Fro gt 1).
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PREDICTIONS OF LINEAR INVISCID SHALLOW-WATER
THEORY FOR DUNES AND ANTIDUNES
23
BEYOND THE SHALLOW-WATER APPROXIMATION POTENTIAL
FLOW FORMULATION
The shallow-water theory of bedforms is not
entirely accurate. This is because the
wavelength ? of dunes and antidunes usually
scales as a multiple of the flow depth H, and so
the condition H/? ltlt 1 is usually not satisfied.
In more precise terms, the wavenumber of the
bedforms k 2?Ho/? does not usually satisfy the
condition k ltlt 1. A better view of bedforms is
obtained by solving for the linearized potential
flow over a wavy bed. This formulation includes
the vertical coordinate z as well as the
horizontal coordinate x, and describes the
vertical as well as the horizontal structure of
the response of the flow to bed. Such a solution
was first implemented by Anderson (1953) and
extended by Kennedy (1963).
Let ? velocity potential function
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POTENTIAL FLOW FORMULATION contd.
In general, subcritical flow is a flow for which
the water surface perturbation is approximately
out of phase with a bed perturbation, and
supercritical flow is a flow for which the water
surface is approximately in phase with a bed
perturbation. Potential flow theory indicates
that the border between subcritical and
supercritical flow is a function of both Froude
number Fro and wavenumber k 2?Ho/? as
follows Now as k ? 0, Fro ? 1, indicating
that for long bedforms the division between
subcritical and supercritical flow is given by
the long wave (shallow-water) limit of 1. If
e.g. the bedform has a wavelength ? equal to 5 Ho
(a reasonable guess for many dunes and
antidunes), k 1.26 and the borderline between
subcritical and supercritical flow is Fro 0.82.
That is, the zone of supercritical response
extends somewhat into the range Fro lt 1, and
antidunes can occur in flows for which Fro lt
1. In general, lower-regime flow refers to truly
subcritical flow in the sense Fro lt
tanh(k)/k1/2, and upper-regime flow refers to
truly supercritical flow in the sense Fro gt
tanh(k)/k1/2. It is important to realize that
part of the zone of upper-regime flow is
subcritical in the long-wave sense.
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POTENTIAL FLOW FORMULATION contd.
In addition to the criterion dividing
subcritical from supercritical response,
potential flow reveals another criterion further
dividing the regime of supercritical flow. When
Fro lt tanh(k)/k1/2 both the water surface and
depth are out of phase with the bed, and the flow
accelerates over bed crests and decelerates over
bed troughs. This gives rise to
downstream-migrating dunes. When Fro gt
tanh(k)/k1/2 and Fro lt k tanh(k)-1/2, both
the water surface and the depth are in phase with
the bed, and the flow decelerates over crests and
accelerates over troughs. This gives rise to
upstream-migrating antidunes. When Fro gt
tanh(k)/k1/2 and Fro gt k tanh(k)-1/2 the
water surface is in phase with the bed, so the
bedforms are antidunes, but the depth is out of
phase with the bed, and the flow accelerates over
the crests and decelerates over the troughs.
These antidunes (which cannot be obtained from
the St. Venant formulation) thus migrate
downstream.
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FLOW IN THE DUNE REGIME
Fro lt tanh(k)/k1/2 Water surface is out of
phase with the bed. Depth variation is out of
phase with the bed Flow accelerates from trough
to crest. Sediment transport increases from
trough to crest. Bedform migrates
downstream. Bedform becomes asymmetric.
27
FLOW IN THE UPSTREAM-MIGRATING ANTIDUNE REGIME
tanh(k)/k1/2 lt Fro lt k tanh(k)-1/2 Water
surface is in phase with the bed. Depth variation
is in phase with the bed Flow decelerates from
trough to crest. Sediment transport decreases
from trough to crest. Bedform migrates upstream
(or hardly at all). Bedform stays symmetric.
28
FLOW IN THE DOWNSTREAM-MIGRATING ANTIDUNE REGIME
k tanh(k)-1/2 lt Fro Water surface is in phase
with the bed. Depth variation is out of phase
with the bed. Flow accelerates from trough to
crest. Sediment transport increases from trough
to crest. Bedform migrates downstream. Bedform
becomes asymmetric. These are antidunes that look
like dunes not too common, but they are observed.
29
PHASE DIAGRAM FOR DUNES AND ANTIDUNES BASED ON
LINEAR POTENTIAL THEORY OVER A WAVY BED
30
LOWER- AND UPPER-REGIME PLANE BED
Dunes usually do not form in gravel-bed streams.
This is because such streams usually fall into a
regime known as lower-regime plane bed, for which
the flow is subcritical and neither dunes nor
ripples form. Chabert and Chauvin (1963) have
described this regime experimentally, and
Engelund (1970) and Fredsoe (1974) have developed
stability analyses for bedforms which describe
this regime. In sand-bed streams, there is a
second regime in the vicinity of the line Fro
tanh(k)/k1/2 within which neither dunes nor
antidunes form. This regime is known as
upper-regime plane bed. Engelund (1970) and
Fredsoe (1974) have explained this region as one
of competition between the effects of bedload and
suspended load. The former favors the formation
of dunes, and the latter favors the formation of
antidunes. Within the regime of upper-regime
plane bed, the two effects cancel each other, and
a plane bed prevails. A rough sketch of the
zones for lower-regime plane bed, dunes, upper
regime plane bed, upstream-migrating antidunes
and downstream-migrating antidunes is given in
the following diagram based on potential flow.
It should be pointed out, however, that the
analyses of Engelund (1970) and Fredsoe (1974)
result in somewhat modified criteria for the
divisions between supcritical and supercritical
flow, and upstream and downstream migrating
antidunes. (See Engelund and Fredsoe, 1982).
31
APPROXIMATE PHASE DIAGRAM FOR 1D BEDFORMS
32
EXPERIMENTAL RESULTS OF CHABERT AND CHAUVIN (1963)
Chabert and Chauvin (1963) report on experiments
which yield a threshold conditions for ripples
that is very similar to that proposed by Engelund
and Hansen (1967). In addition, they obtain a
criterion for the threshold between lower-regime
plane bed and dunes that can be approximated as
where ?c is given by the modified Brownlie
relation, Thus in the limit of coarse material
(Rep gtgt 1, gravel-bed streams) dunes should not
form until ? exceeds 0.0816. It was seen from
Slide 21 of Chapter 3, however, that this
condition is not common for gravel-bed streams at
bankfull flow. Dunes can form in gravel-bed
streams if the conditions are right e.g. see
Dinehart (1992).
33
SHIELDS DIAGRAM INCLUDING RESULTS OF CHABERT AND
CHAUVIN (1963)
34
CYCLIC STEPS (CHUTE-AND-POOL TOPOGRAPHY)
Trains of cyclic steps occur in very steep flows
with supercritical Froude numbers. They are
long-wave relatives of antidunes (Winterwerp et
al., 1992 Taki and Parker, in press). The steps
are delineated by hydraulic jumps (immediately
downstream of which the flow is locally
subcritical). The steps migrate upstream. These
features are also called chute-and-pool
topography (Simons et al., 1965). Their regime
of formation is schematized in the previous Slide
30.
Train of cyclic steps in a small laboratory
channel at St. Anthony Falls Laboratory. The
water has been dyed to aid visualization two
hydraulic jumps can be seen in the figure.
35
1D BEDFORM REGIME DIAGRAMS
A number of diagrams have been proposed to
characterize bedform regime. The most useful of
these are dimensionless. Following the analysis
of Vanoni (1974) and Parker and Anderson (1978),
the following general relation can be posited
Here Rep, R and ?g have their standard
meanings of explicit particle Reynolds number,
sediment submerged specific gravity and geometric
standard deviation of bed sediment. In addition,
X1 and X2 are two dimensionless parameters
describing the flow which must be independent
from each other. Suitable choices include
Shields number ? u2/(RgDs50), Froude number
Fr qw/(gH)1/2H, bed slope S, dimensionless
depth H/Ds50, dimensionless unit stream
power US/vs etc. Vanoni (1974) has provided a
relatively complete set of bedform diagrams for
sand, including dunes, antidunes, ripples, flat
(by which he means lower-regime flat bed),
transition (by which he means upper-regime flat
bed) and chute-and-pool topography (in which
cyclic steps should be included. Vanoni chooses
X1 Fr and X2 , In addition, he assumes
that R is constant at 1.65, and he neglects the
effect of ?g (i.e. assumes uniform material), so
that
36
BEDFORM REGIME DIAGRAM 1 OF VANONI (1974)
Fr
Bedform Chart for D50 0.011 mm and 0.088 0.15
mm (Rep 0.11 and 2.4 5.4)
37
BEDFORM REGIME DIAGRAM 2 OF VANONI (1974)
Fr
Bedform Chart for D50 0.12 0.20 mm (Rep
3.9 8.3)
38
BEDFORM REGIME DIAGRAM 3 OF VANONI (1974)
Fr
Bedform Chart for D50 0.15 0.32 mm (Rep
5.4 16.8)
39
BEDFORM REGIME DIAGRAM 4 OF VANONI (1974)
Fr
Bedform Chart for D50 0.23 0.45 mm (Rep
10.2 28.0)
40
BEDFORM REGIME DIAGRAM 5 OF VANONI (1974)
Fr
Bedform Chart for D50 0.4 0.6 mm (Rep 23.5
43.1)
41
BEDFORM REGIME DIAGRAM 6 OF VANONI (1974)
1.0
Fr
0.1
103
102
Bedform Chart for D50 0.93, 1.20 and 1.35 mm
(Rep 83.3, 122, 146)
42
BEDFORM REGIME DIAGRAM OF ENGELUND AND HANSEN
(1966)
This diagram uses the hydraulic parameters X1
Fr and X2 U/u. The parameter Rep is not
included, and the diagram is valid only for
sand. The diagram clearly shows an extensive
range of flow for which Fr lt 1 but antidunes
form. The plane bed regime on the left-hand
side of the diagram is upper-regime plane bed.
Lower-regime plane bed is not shown in the
diagram.
U/u
Fr
43
REFERENCES FOR CHAPTER 8
Anderson, A. G., 1953, The characteristics of
sediment waves formed by flow in open channels,
Proceedings, 3rd Midwest Conference on Fluid
Mechanics, University of Minnesota. Chabert, J.
and Chauvin, J. L., 1963, Formation des dunes et
de rides dans les modeles fluviaux, Bulletin
C.R.E.C., No. 4. Dinehart, R. L., 1992, Evolution
of coarse gravel bed forms field measurements at
flood stage, Water Res. Res., 28(10), 2667-2689.
Engelund, F., 1970, Instability of erodible
beds, J. Fluid Mech., 42(2). Engelund, F. and
Fredsoe, J., 1982, Sediment ripples and dunes,
Annual Review of Fluid Mechanics, 14, 13-37.
Engelund, F. and Hansen, E., 1966, Hydraulic
resistance in alluvial streams, Acta Polytechnica
Scandanavica, V. Ci-35. Engelund, F. and Hansen,
E., 1967, A Monograph on Sediment Transport,
Technisk Forlag, Copenhagen, Denmark. Fredsoe,
J., 1974, On the development of dunes in erodible
channels, J. Fluid Mech., 64(1), 1-16. Kennedy,
J. F., 1963, The mechanics of dunes and antidunes
in erodible bed channels, J, Fluid Mech., 16(4).
Parker, G. and Anderson, A., 1977, Basic
principles of river hydraulics, J. Hydr. Engrg.,
103(9), 1077-1087.
44
REFERENCES FOR CHAPTER 8 contd.
Simons, D. B., Richardson, E. V. and Nordin, C.
F., 1965, Sedimentary structures generated
by flow in alluvial channels, Special Pub. No.
12, Am. Assoc. Petrol. Geologists. Taki, K.. And
Parker, G., 2005, Transportational cyclic steps
created by flow over an erodible bed. Part 1.
Experiments, J. Hydr. Res., in press,
downloadable from http//cee.uiuc.edu/people
/parkerg/preprints.htm . Vanoni, V., 1974,
Factors determining bed forms of alluvial
streams, Journal of Hydraulic Engineering,
100(3), 363-377. Winterwerp, J. C., Bakker, W.
T., Mastbergen, D. R., and Van Rossum, H, 1992,
Hyperconcentrated sand-water mixture flows over
erodible bed, J. Hydr. Engrg., 119(11), 1508-1525.